Mathematical Methods I, Autumn 2009
61
V (x, t) = E V (Xt+dt , t + dt) | Xt = x .
(5.7)
so that
Next, by Itos lemma (V can be shown to be twice continuously dierentiable),
V
1 2V
(Xt , t) dXt +
(Xt , t)(dXt )2
2
x
2
x
V
(Xt , t)2 2 V
(Xt , t) +
= V (Xt )
Mathematical Methods I, Autumn 2009
67
where we used that dierent components of (Zt )t0 are independent at all times
(possibly dierent for the dierent components). Finally, the third and fourth
properties are immediate consequences of similar properties o
Mathematical Methods I, Autumn 2009
69
Theorem 6.3 (It
os lemma for vector processes). Let Xt = (X1,t , . . . , Xn,t )
be a vector It
o process dened by (6.11). Then
df (Xt , t) =
f
f
1 2f
dt +
dXj,t +
dXj,t dXk,t ,
t
xj
2
xj xk
j
(6.14)
j,k
which can al
Mathematical Methods I, Autumn 2009
=
since
p
ip jp = i2
73
1
j i2 dt +
ij dZj,t ,
2
j
hip hpi = i2 ii = i2 . From there, proceed as before.
Example 6.8 (Weakly coupled Ornstein - Uehlenbeck). The following
example is a 2-dimensional version of the Orn
Mathematical Methods I, Autumn 2009
77
and deduce form this that Yt as variance-covariance matrix
t
T
E Yt Yt =
exp u1 A exp uT AT T ,
0
where T stands for the adjoint of 1 .
Hint. Use
that
the variance-covariance matrix of the stochastic dieren
Mathematical Methods I, Autumn 2009
63
and, for a general European pay-o of f (ST ) at T ,
V (S, T ) = f (S),
(5.12)
f being specied by the derivatives contract.
Now (5.11)(5.12) is precisely the type of problem to which we can apply
our baby FeynmanKac t
Chapter 5
Stochastic processes and
PDEs
There is a close connection between solving an sde and solving boundary value
problems for a certain type of partial dierential equations (pdes) which are
known as parabolic. Parabolic pdes are, roughly speaking, th
Mathematical Methods I, Autumn 2009
71
(Here Ht is not the H from the decomposition (6.3) of .) Since dWp,t dWq,t =
p,q , we nd that df (Xt , t) equals
f
2f
f
f
1
+
aj,t
+
pq hjp,t hkq,t
hjp,t
dWp,t ,
dt +
t
x
2
x
x
x
j
j k
j
p,q
p
j
j
j,k
Note that wit
Chapter 6
Multivariable Ito Calculus
As is the case for ordinary Calculus, there exists a multi-variable version of
Ito Calculus involving more than one Brownian. It is relevant for modelling
situations in which there are several independent sources of un
1/11/06 6:22 PM C:\Dan\CSU\Book\Problems\Particle\Kernel.m 2 of 2
_
for i E 2 : NReg — 1
xreg(i) = xreg(i—1) + dx;
end
% Create the pdf approximation that is required for the regularized
% particle filter.
for j = 1 : NReg
for i = 1 : N
qreg(i.j) = 0:
nor
1/11/06 6:18 PM C:\Dan\CSU\Book\Problems\Partic1e\ngersghere.m 1 of 1
function Hypersphere
% Optimal State Estimation Solution Manual, by Dan Simon
% Problem 15.11
% Plot the volume of a hypersphere as a function of dimension.
v(1) = 2;
V(2) = pi;
i 1 : 2
FIGURE
9
Network Representation
of Equipment
Replacement
Time
0
1
2
3
4
5
An Alternative Recursion
There is another dynamic programming formulation of the equipment-replacement model.
If we dene the stage to be the time t and the state at any stage to be
incurred at the end of month 2. Thus, the total cost incurred during month 2 is (1)(i
2
x 3) c(x). During months 3 and 4, we follow an optimal policy. Since month 3 begins with an inventory of i x 3, the cost incurred during months 3 and 4 is f3(i
x 3).
cj
In terms of benet per unit weight, the best item is the item with the largest value of .
wj
Assume there are n types of items that have been ordered, so that
c1
c2
cn
w1
w2
wn
Thus, Type 1 items are the best, Type 2 items are the second best, and so
This often occurs when the stage alone supplies sufcient information to make an optimal decision. We now work through several examples that illustrate the art of formulating dynamic programming recursions.
EXAMPLE
8
A Fishery
The owner of a lake must deci
FIGURE
8
Time Horizon for
Equipment
Replacement
Year 1
Time
0
Year 2
Time
1
Year 3
Time
2
Year 4
Time
3
Year 5
Time
4
Time
5
1, 2, or 3 years; after i years of use (i 1, 2, 3), it may be traded in for a new one. If an
i-year-old engine analyzer is traded
terminate our computations by computing f1(6). Then we retrace our steps and determine
the amount that should be allocated to each investment (just as we retraced our steps to
determine the optimal production level for each month in Example 4).
Stage 3 Co
$
$
$
Th
sh is
ar stu
ed d
vi y re
aC s
ou
ou rc
rs e
eH w
er as
o.
co
m
!
https:/www.coursehero.com/file/7193411/25BIS-2C-Midterm-1/
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Bis 2c winter midterm 1 answer key
SA1
C1, C2, C4,C5
C1 and C4
C2 and
Problem 2 (10 Points)
Lamborghini, an Italian sports cars maker, asked you to design a relational database that keeps track of its
part suppliers and the purchases of parts by its departments from these suppliers.
A part is identified by it its PID number
Chemistry 14D Lecture 1
Fall 2012
Final Exam Part A
Page 3
10. (15) Write the major product(s) for each reaction shown below. If no reaction occurs, write "NR". Assume the
reactants above and below the arrow are present in excess. Do not provide any mecha
Chemistry 14D Lecture 1
Br
Fall 2012
Final Exam Part B Solutions
Page 3
Br
Br
29.
Ph
Free radical chain mechanisms end with a propagation step,
not a termination step.
+ Br
Ph
30. Atom transfer (2nd and 3rd mechanism steps), initiation (1st mechanism step
Total transaction costs=$92,680.
e. What if Grant Hill Associates cash holdings of Asia money are 15 billion Indonesia rupiahs, 18 million
Malaysia ringgits, and 2.4 billion yens, it is not clear how these holdings should be converted back into
dollars, w
Chemistry 14D Lecture 1
Fall 2012
Final Exam Part B Solutions
Page 1
Statistics: High score, average and low score will be posted on the course web site after exam grading is complete.
A note about exam keys: The answers presented here may be significantl